Quotient group

A quotient group is a group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure.

Definitions

Let $\text{[math]}$ be a group and $\text{[math]}$ a normal subgroup.

Definition via cosets

The quotient group $\text{[math]}$ is the set of left cosets

$\text{[math]}$

with group operation $\text{[math]}$

Definition via equivalence relations

Define a relation on $\text{[math]}$ by

$\text{[math]}$

The relation $\text{[math]}$ is a equivalence relation, and denote $\text{[math]}$ as the equivalence class of $\text{[math]}$ . The quotient group is defined by the set of equivalence classes

$\text{[math]}$

with operation $\text{[math]}$